From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1804 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: Michael Healy's question on math and AI Date: Wed, 24 Jan 2001 23:26:27 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed X-Trace: ger.gmane.org 1241018116 562 80.91.229.2 (29 Apr 2009 15:15:16 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:15:16 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Jan 25 14:38:55 2001 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f0PHsuw13460 for categories-list; Thu, 25 Jan 2001 13:54:56 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Originating-IP: [148.235.239.152] X-OriginalArrivalTime: 24 Jan 2001 23:26:27.0993 (UTC) FILETIME=[1347D890:01C0865D] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 35 Original-Lines: 119 Xref: news.gmane.org gmane.science.mathematics.categories:1804 Archived-At: Re : Michael Healy’s question on math and AI This is to answer Mike and also several other people who have contacted me recently asking how I would respond to queries about (1) Artificial Intelligence, cognitive science, linguistic engineering, knowledge representation, and related attempts at creating modern subjects, and (2) the relevance of category theory and of mathematics in general to these. My basic response is strong advice to actually learn some category theory, rather than resting content with slinging back and forth ill-defined epithets like “set theory”, “contingency”, etc.. So much confusion has been accumulated that an opposition of the form “set-theoretical versus non-set-theoretical” has at least seven wholly distinct meanings, hence billions of electrons and drops of ink can be spilled by surreptitiously identifying any two of these. For example, the opposition can concern whether or not large cardinal assumptions are needed for a certain result, which is mathematically meaningful and hence independent of whether or not the ZFvN rigidification of Cantor is being used as a framework. Another example is the opposition habitually used in geometry between properties of spaces which can be explained in terms of arbitrary mappings versus those which depend on the cohesion being studied (e.g. “the underlying abstract group vs. the Lie group”). Obviously these two oppositions are not the same although they may be related. One of the oppositions which I have emphasized since 1964 is the ZFvN rigid hierarchy based on galactically “meaningful” inclusion, requiring the totally arbitrary “singleton” operation of Peano with the resulting chains of mathematically spurious rigidified membership, on the one hand, versus the category of abstract sets, involving many potential universes of discourse and arbitrary specific relations between them, on the other hand. (Abstract sets can CARRY structures of mathematical interest, but precisely because of the need of flexibility in the latter, they themselves have only very few properties, unlike the ZFvN “sets”). Within Cantor’s original conception itself there is a fundamentally important opposition: the abstract sets, which he called “Kardinalzahlen”, versus the cohesive and variable sets which he called “Mengen”. (An additional confusion stems from the use, by nearly all of Cantor’s followers, of the term “cardinal number” to mean (not a Kardinalzahl=abstract set, but) a label for an isomorphism class of abstract sets, an invariant which Cantor of course also studied, but which is too abstract to support the specific relations between abstract sets themselves, the mappings, and hence cannot carry the needed mathematical structures). (A) The real issue is that for purposes of pure AND applied mathematics, we need to be able to represent (without spurious ingredients) these cohesive and variable sets (or “spaces”) and their relationships. The ZFvN rigidification fails so miserably in doing this that even those geometers and analysts who pay lip service to it as a “foundation” never in practice use its formalism. (B) Category theory made explicit some universal features of the relationship between quantity and quality whose fundamental importance had been forced into consciousness by the work of Volterra and Hurewicz (both of whom made basic contributions to both functional analysis and algebraic topology) and of many others. This relationship between quantitative and qualitative aspects concerns cohesive and variable sets and structures built on such spaces. For example, Volterra already recognized that spaces have “elements” other than points, and Hurewicz recognized the need for cartesian-closed categories (even before the lambda-calculus formalism, or category theory, was devised); moreover, the original fiber bundles were explicitly modeling dynamical situations, etc. Many people working in the new fields, striving to realize the dream of a theoretical computer science, do not seem to be aware of points like (A) and (B). It would certainly be a bad strategy for the advancement of science to “hide” the fact that category theory belongs to the background of a new result and thus to help perpetuate that sort of ignorance. The role of mathematics in general (not only of category theory) also seems to be widely misunderstood, even in those fields which definitely need more mathematics in order to mature and make a real contribution. For example, some say that logic is more general than mathematics, partly because of ignoring the strongly qualitative aspect of modern mathematics and partly because of the philosophical tradition of hiding the fact that no logic other than mathematical logic has had any significant real-world applications. Because of the minimal mathematical education required of students of philosophy, the claim is too easily accepted in many philosophical circles that “mathematics is unsuitable” for some given issue of conceptual analysis; this conclusion seems to be based on the syllogism: mathematics is set theory (a misconception which the philosophers themselves have done much to disseminate), set theory is clearly not suitable (actually because of the defects of the ZFvN rigidification, which make it ill-suited for mathematics as well) hence ...... This syllogism serves as an excuse to indefinitely postpone learning mathematics (and category theory in particular). An older sort of excuse is the assertion that the proposed science should concern the REAL WORLD, not pure mathematics. This superficially appealing truism has frequently been used to mask the fact that comparing reality with existing concepts does not alone suffice to produce the level of understanding required to change the world; a capacity for constructing flexible yet reliable SYSTEMS of concepts is needed to guide the process. This realization (not Platonism) was the basis of the supreme respect for mathematics expressed by champions of reality like Galileo, Maxwell, and Heaviside. For example, the differential calculus provides the capacity to construct systems descriptive of celestial motions, fluid interactions, electromagnetic radiation fields, etc., and therefore engineers have learned it. The functorial calculus helps to provide a similar capacity adequate to the requirements, not only of the older sciences, but of the newer would-be sciences as well. Hence my response. Bill Lawvere